The Game of Life in Color

Auxilary materials for "One Tub, Eight Blocks, Twelve Blinkers and Other Views of Life", by John Pulsifer and Cliff Reiter [3]

The Game of Life is a well known and intriguing process that was first described by John Conway[1-2]. It is a two dimensional automaton that operates on square arrays of cells. Each cell has a value of 0 (dead) or 1 (alive). Cells which are alive at one generation are alive at the next if 2 or 3 of their 8 neighbors are alive; cells that are dead at one generation are alive at the next if exactly 3 of their 8 neighbors are alive. This automaton is intriguing because complex behaviors occur. Periodic and moving structures appear and visual representations contain both suggestions of structure and randomness. It is traditional to view the evolution of the automaton via black and white animations. This note follows the convention in "One Tub, Eight Blocks, Twelve Blinkers and Other Views of Life" [3] and marks dead cells as white (or transparent) and alive cells are marked according to the number of neighbors. Indeed, this note is meant to augment that note. The following figure shows our convention for color.
Following [3], we consider some examples which illustrate the richness of the automaton, its behavior on squares, and behavior on random configurations. We will use both 3-d representations and animations to visualize the time evolution of this automaton.

Some first views

 Some periodic patterns and a glider (3d) A glider gun (3d) The B-heptomino where a very small structure (7 living cells) evolves in a complex way for over a hundred generations before simplifying Some periodic patterns and a glider (animation) A glider gun (animations) The B-heptomino (animation)

A study of the evolution of square configurations

Given the symmetry of the initial configurations and the rules, each of these examples maintain 4-fold symmetry. These configurations are simplier and more symmetric than random configurations.
 A 12 by 12 square (3d) A 13 by 13 square (3d) A 14 by 14 square (3d) A 12 by 12 square (animation) A 13 by 13 square (animation) A 14 by 14 square (animation) A 15 by 15 square (3d) A 16 by 16 square (3d) A 17 by 17 square (3d) A 15 by 15 square (animation) A 16 by 16 square (animation) A 17 by 17 square (animation) A 18 by 18 square (3d) A 19 by 19 square (3d) A 20 by 20 square (3d) A 18 by 18 square (animation) A 19 by 19 square (animation) A 20 by 20 square (animation) A 21 by 21 square (3d) A 22 by 22 square (3d) A 23 by 23 square (3d) A 21 by 21 square (animation) A 22 by 22 square (animation) A 23 by 23 square (animation)

Behavior on random configurations

Here we see the evolution on random configuration, with periodic boundary conditions, may take hundreds of generations to simplify.
 Random configuration I; 400 step animation Random configuration II; 600 step animation Random configuration III; 1000 step animation

References
1. E. Berlekamp, J. Conway, and R. Guy: Winning Ways For Your Mathematical Plays. Academic Press, New York, 1982.
2. M. Gardner: The fantastic combinations of John Conway's new solitaire game of "life". Scientific American, 223 4, pp 120-123, 1970.
3. John E. Pulsifer and Clifford A. Reiter, One Tub, Eight Blocks, Twelve Blinkers and Other Views of Life, Computers & Graphics, 20 3(1996) 457-462. [Abstract]
4. A paper looking at fuzzy automata, including the Game of Life:
Fuzzy Automata and Life, [Images, and Animations], [abstract]
5. There are many great web pages on The Game of Life; some good starting points are:
Conway's Game of Life, Al Hemsel, 2001, http://hensel.lifepatterns.net/
6. Patterns, Programs, and Links for Conway's Game of Life, Paul Callahan, http://www.radicaleye.com/lifepage/lifepage.html#catback